Claim Summary

Section-by-section structural enumeration with epistemic-status labels.

1 (The Problem). SYNTHESIS. The gauge programme so far. Gauge SU(2) × U(1)_Y (AP27), U(1) (AP15), electroweak unification (AP16). SU(3) is the last piece.

2 (Three Expressions of One Manifold). DERIVED. N = 3 spatial dimensions from {C, S, B} (AP10). Each axiom expresses one face. Functional identity fixed (the axiom names the function); geometric assignment free (the axiom carries no spatial tag). Linked but non-derivable (AP20 §4.2, fenced KS-P.2).

3 (The Silent Pop). DERIVED / SYNTHESIS. §3.1–3.2: the break creates direction as a concept; the manifold orients around the splinter (DERIVED). §3.3: endless possibilities across many actualisations (SYNTHESIS, non-load-bearing). §3.4: two views of the same now — before collapse, all orientations coexist in ℂ³ superposition; after collapse, record history selects one (DERIVED, AP09).

4 (The Gauge Freedom). DERIVED. §4.1: physics does not depend on orientation; non-derivability of {C, S, B} → global symmetry; per-event orientation → local symmetry → gauge theory. §4.2: Lemma (orientation ∈ ℂ³ from AP09); Proposition 1 (gauge group is SU(3) — norm-preserving on ℂ³ is U(3); factor out the orientation’s global phase (Born-rule redundancy, AP09); SU(3) with 8 generators).

5 (Colour). DERIVED / STRUCTURAL. §5.1: colour = orientation, anticolour = conjugate representation (DERIVED). §5.2: Proposition 2 (colour neutrality from isotropy + AP15 stiffness) STRUCTURAL — the energy functional is not yet computed (D1). §5.3: gluons as 8 gauge bosons of SU(3) (DERIVED). §5.4: Proposition 4 (ε = electron is SU(3) singlet because minimum break is necessarily isotropic) DERIVED.

6 (Confinement). STRUCTURAL. §6.1: confinement from substrate restoring isotropy (qualitative — D1). §6.2: asymptotic freedom from the same mechanism at short range (qualitative — D3). §6.3: Proposition 3 (SU(3) is exact) DERIVED — Axiom B contains no clause selecting orientation.

7 (The Full Gauge Structure). DERIVED. SU(3) × SU(2) × U(1) assembled. Lie algebra structure su(3) ⊕ su(2) ⊕ u(1). Breaking pattern: SU(3) exact, gluons massless; SU(2) × U(1) broken to U(1)_em by Axiom B (AP16). Global structure (SU(3) × SU(2) × U(1))/ℤ₆ held open in §0.7.

8 (Derivation Chain). SYNTHESIS. Step-by-step from axioms to SU(3) enumerated.

9 (Three Generations). CONJECTURE. ε with three orientations identified as a structural site for the three generations. Mass ratios not computed (D2). Colour vs generation disambiguated: same three faces, two different degrees of freedom (continuous SU(3) orientation vs structural dominance index).

10 (Kill Switches). REGISTER. KS-48c, KS-49b, KS-50 in Claim/Test/Status/Recovery form. Identifiers per the Master Kill Switch Registry; the c/b suffixes resolve collision with AP23 (KS-48a, KS-48b, KS-49a).